nLab iterated algebraic K-theory

Idea

General

The construction of algebraic K-theory $K(R)$, originally defined for rings $R$, generalizes to $A_\infty$-ring spectra. When $R$ happens to be a connective $E_\infty$-ring spectrum, then also the representing spectrum $K(R)$ of its algebraic K-theory is a connective $E_\infty$-ring spectrum (Schwänzl & Vogt 1994, Thm. 1, EKMM 1997, Thm. 6.1) so that this construction may then be iterated (Rognes 2014) to yield iterated algebraic K-theories $K(K(R))$, $K(K(K(R)))$, etc.

The red-shift conjecture says that this iteration plays a special role in chromatic homotopy theory.

On topological K-theory

The construction of iterated algebraic K-theory has received particular attention for the case that $R =$ ku is the connective ring spectrum representing complex topological K-theory.

Here the first iterated stage $K(ku)$ is related to BDR 2-vector bundles essentially like ku is related to ordinary complex vector bundles.

The tower $K^{2r}(ku)$ of higher iterated algebraic K-theories of topological K-theory has been shown to accommodate a generalization of the Fourier-Mukai-type transform on twisted K-theory that is given by topological T-duality, generalizing it to spherical T-duality (Lind-Sati-Westerland 16).

Properties

Red-shift conjecture

See at red-shift conjecture.

References

Algebraic K-theory of ring spectra

On the algebraic K-theory of (connective) ring spectra:

• Roland Schwänzl, Rainer Vogt, Basic Constructions in the K-Theory of Homotopy Ring Spaces, Transactions of the American Mathematical Society, Vol. 341, No. 2 (Feb., 1994), pp. 549-584 (jstor:2154572, doi:10.2307/2154572)

• Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, chapter VI of Rings, modules and algebras in stable homotopy theory, AMS Mathematical Surveys and Monographs Volume 47 (1997) (pdf)

• Andrew Blumberg, David Gepner, Gonçalo Tabuada, Section 9.5 of: A universal characterization of higher algebraic K-theory, Geom. Topol. 17 (2013) 733-838 (arXiv:1001.2282, doi:10.2140/gt.2013.17.733)

• Jacob Lurie, Algebraic K-Theory of Ring Spectra, Lecture 19 of Algebraic K-Theory and Manifold Topology, 2014 (pdf)

The algebraic K-theory specifically of suspension spectra of loop spaces (Waldhausen’s A-theory) is originally due to

• Friedhelm Waldhausen: Algebraic K-theory of spaces, in: Algebraic and Geometric Topology (Proceedings of a conference at Rutgers, New Brunswick, N. J., 1983), Lecture Notes in Math. 1126, Springer (1985) 318-419 [doi:10.1007/BFb0074449, pdf, pdf]

On the iteration of the construction and the red-shift conjecture:

• John Rognes, Chromatic redshift, lecture notes, MSRI 2014 (arXiv:1403.4838)

On the algebraic K-theory $K(R)$ of a ring spectrum $R$ as the Grothendieck group of (∞,1)-module bundles over $R$:

• John Lind, Bundles of spectra and algebraic K-theory, Pacific J. Math. 285 (2016) 427-452 (arXiv:1304.5676, doi:10.2140/pjm.2016.285.427)

Algebraic K-theory of topological K-theory

On the first algebraic K-theory $K(ku)$ of connective topological K-theory:

• Christian Ausoni, On the algebraic K-theory of the complex K-theory spectrum, Inventiones mathematicae volume 180, pages 611–668 (2010) (arXiv:0902.2334, doi:10.1007/s00222-010-0239-x)

• Christian Ausoni, John Rognes, Algebraic K-theory of topological K-theory, Acta Math. Volume 188, Number 1 (2002), 1-39 (euclid:acta/1485891473)

• Christian Ausoni, John Rognes, Rational algebraic K-theory of topological K-theory, Geom. Topol. 16 (2012) 2037-2065 (arXiv:0708.2160, doi:10.2140/gt.2012.16.2037)

Interpretation of $K(ku)$ as the K-theory of BDR 2-vector bundles:

• Nils Baas, Bjørn Ian Dundas, John Rognes, Two-vector bundles and forms of elliptic cohomology, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004 (arXiv:math/0306027, doi:10.1017/CBO9780511526398.005)

• Nils Baas, Bjørn Ian Dundas, Birgit Richter, John Rognes, Stable bundles over rig categories, Journal of Topology, Volume 4, Issue 3, September 2011, Pages 623–640 (arXiv:0909.1742, doi:10.1112/jtopol/jtr016)

which can also be understood as a special case of K-theory for 2-categories:

• Nick Gurski, Niles Johnson, Angélica M. Osorno. K-theory for 2-categories. Advances in Mathematics 322 (2017) 378-472 [doi, arXiv:1503.07824]

Algebraic K-theory of algebraic K-theory

On the algebraic K-theory of algebraic K-theory of finite fields $K(K(\mathbb{F}))$:

• Gabe Angelini-Knoll, Detecting the $\beta$-family in iterated algebraic K-theory of finite fields, Wayne State University 2017 (arXiv:1810.10088, oa_dissertations:1778)

Higher iterated algebraic K-theory of topological K-theory

Discussion of higher and of twisted iterated K-theory on $ku$, and realization of the spherical T-duality on twisted $K^{2r}(ku)$:

• John Lind, Hisham Sati, Craig Westerland, A higher categorical analogue of topological T-duality for sphere bundles, Annals of K-Theory, Vol. 5 (2020), No. 1, 1–42 (arXiv:1601.06285, doi:doi:10.2140/akt.2020.5.1)